In general, a web handling or roll-to-roll system refers to a system in which a web of a material having a width and length significantly larger than thickness, such as a plastic film or a thick iron sheet material, passes through rolls, while it is continuously subjected to various processes.
Among the production sections of the web handling system, the winding section is an important process. A process for producing center-wound rolls has advantages in that it is efficient, provides a large storage space and is very convenient in high-speed operations. However, the non-uniform stresses within the rolls can cause damages such as buckling, spoking, cinching, etc. For this reason, a winding process, which avoids the occurrence of excessive or unnecessary internal stress and induces stable stress distribution, is required.
With respect to prior papers, Altmann presented a general solution for a linear elastic roll material while using a nonlinear constitutive relation to find the radial and hoop stresses for successive wraps [4]. In addition, Altmann proposed a second-order differential equation for the linear elastic material in a center-wound roll.
Yagoda established the core compliance as an inner boundary condition on center-wound rolls [5], and Hakiel incorporated nonlinear material properties into the basic mechanics and numerical solutions of wound roll stresses [3].
Good compared results from Hakiel s model with interlayer pressure measurements obtained using pull tabs [2].
They noted that the model typically predicted stresses that were twice as large as their measured values. However, they were able to bring predicted and measured values into better agreement by modifying the outer hoop-stress boundary condition to relax relative to the out-layer tensile stresses by their model of “wound on tension” loss.
Burns et al. derived a strain-based formula for stresses in profiled centre wound rolls by using a residual stress model [1]. They noted that radial stress within wound rolls is closely related to the variation of effective residual stress.
The present inventors have found that a momentous factor for making a high quality wound roll is the taper tension profile of the winding process. Also, in the present invention, an auto taper tension profile making method for avoiding the damage (telescoping, buckling, cinching, etc.) is presented. The experimental results revealed that the proposed method is very useful.
FIG. 1 is a schematic diagram of the tension T acting on the web and roll. In FIG. 1, “a” is a core radius, “R” is the current radius of the roll, “M” is torque, and “δω” is a taper tension profile.
In general, a linear taper tension profile and a hyperbolic taper tension profile are applied to winding processes [2][3]. Herein, the linear taper tension profile is a profile in which tension linearly decreases with an increase in the radius of the roll, and the hyperbolic taper tension profile is a profile in which tension hyperbolically decreases with an increase in the radius of the roll.
The linear and hyperbolic taper tension profiles are represented by the following Math Figures 1 and 2, wherein “σ0” is initial web stress, taper is the decrement for taper tension, and r is dimensionless roll radius ratio, i.e., the value obtained by dividing the roll radius by the core radius:
                                          σ            w                    ⁡                      (            r            )                          =                              σ            0                    ⁡                      [                          1              -                                                (                                      taper                    100                                    )                                ⁢                                                      (                                          r                      -                      1                                        )                                                        (                                          R                      -                      1                                        )                                                                        ]                                              [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            1                          ]                                                      σ            w                    ⁡                      (            r            )                          =                              σ            0                    ⁡                      [                          1              -                                                (                                      taper                    100                                    )                                ⁢                                  (                                                            r                      -                      1                                        r                                    )                                                      ]                                              [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            2                          ]            
FIG. 2 shows the taper tension plotted as a taper tension ratio, i.e., (σw(r)/σ0), for the two profiles. The hyperbolic taper tension variation is larger at the core and smaller toward the outer layer, but the linear taper tension variation is constant.
The boundary condition is that the outside of the roll is stress free. Thus, stress for the radial direction within the wound roll is given in Math Figure 3 [1].
                              σ          rr                =                              1            r                    ⁢                      {                                          [                                  B                  (                                                            r                      β                                        -                                                                  R                                                  2                          ⁢                          β                                                                                            r                        β                                                                              )                                ]                            +                                                1                                      2                    ⁢                    β                                                  ⁡                                  [                                                                                                                                                                        r                                                              -                                β                                                                                      ⁢                                                                                          ∫                                r                                R                                                            ⁢                                                                                                t                                  β                                                                ⁢                                                                                                      σ                                    *                                                                    ⁡                                                                      (                                    t                                    )                                                                                                  ⁢                                                                  ⅆ                                  t                                                                                                                                              -                                                                                                                                                                                          r                            β                                                    ⁢                                                                                    ∫                              r                              R                                                        ⁢                                                                                          t                                                                  -                                  β                                                                                            ⁢                                                                                                σ                                  *                                                                ⁡                                                                  (                                  t                                  )                                                                                            ⁢                                                              ⅆ                                t                                                                                                                                                                                          ]                                                      }                                              [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            3                          ]            
In Math Figure 3,
                              B          =                                                                                                                2                      ⁢                      β                      ⁢                                                                                          ⁢                                              σ                        0                                            ⁢                                              E                        c                                            ⁢                                              s                        22                                                              -                                          {                                                                        [                                                                                                                    E                                c                                                            ⁡                                                              (                                                                                                      s                                    12                                                                    -                                                                      β                                    ⁢                                                                                                                                                  ⁢                                                                          s                                      22                                                                                                                                      )                                                                                      -                            1                                                    ]                                                ⁢                                                                              ∫                            1                            R                                                    ⁢                                                                                    t                              β                                                        ⁢                                                                                          σ                                *                                                            ⁡                                                              (                                t                                )                                                                                      ⁢                                                          ⅆ                              t                                                                                                                          }                                        -                                                                                                                    {                                                                  [                                                                                                            E                              c                                                        ⁡                                                          (                                                                                                s                                  12                                                                -                                                                  β                                  ⁢                                                                                                                                          ⁢                                                                      s                                    22                                                                                                                              )                                                                                -                          1                                                ]                                            ⁢                                                                        ∫                          1                          R                                                ⁢                                                                              t                                                          -                              β                                                                                ⁢                                                                                    σ                              *                                                        ⁡                                                          (                              t                              )                                                                                ⁢                                                      ⅆ                            t                                                                                                                }                                                                                      2              ⁢                              β                ⁡                                  [                                                                                    (                                                                                                            s                              12                                                        ⁢                                                          E                              c                                                                                -                          1                                                )                                            ⁢                                              (                                                  1                          -                                                      R                                                          2                              ⁢                              β                                                                                                      )                                                              +                                          β                      ⁢                                                                                          ⁢                                              E                        c                                            ⁢                                                                        s                          22                                                ⁡                                                  (                                                      1                            +                                                          R                                                              2                                ⁢                                β                                                                                                              )                                                                                                      ]                                                                    ⁢                                  ⁢        and                            [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            4                          ]                                          β          2                =                                                            s                11                            ⁢                              s                33                                      -                          s              13              2                                                                          s                22                            ⁢                              s                33                                      -                          s              23              2                                                          [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            5                          ]            
In Math Figures 3 and 4, Ec is the hub core stiffness, and S11, S13, S22, S23 and S33 are the roll's elastic compliances. Substituting the ERS into Math Figure 3 results in Math Figure 6, which means the radial stress for the linear taper tension profile, and the radial stress for the hyperbolic taper tension profile is represented by Math Equation 7:
                                                                        σ                rr                            =                            ⁢                                                1                  r                                ⁢                                  {                                                            [                                              B                        (                                                                              r                            B                                                    -                                                                                    R                                                              2                                ⁢                                β                                                                                                                    r                              β                                                                                                      )                                            ]                                        +                                                                  (                                                  1                                                      2                            ⁢                            β                                                                          )                                            ⁢                                              (                                                                              σ                            0                                                                                1                            -                            v                                                                          )                                                                                                                                                                                  ⁢                              [                                                                            (                                                                                                    R                                                          β                              +                              1                                                                                -                                                      r                                                          β                              +                              1                                                                                                                                β                          +                          1                                                                    )                                        ⁢                                          r                                              -                        β                                                                              +                                                            (                                                                                                    R                                                          1                              -                              β                                                                                -                                                      r                                                          1                              -                              β                                                                                                                                β                          -                          1                                                                    )                                        ⁢                                          r                      β                                                                      ]                                                                                                      ⁢                              {                                                                            (                                                                        2                          +                          v                                                                          1                          +                          v                                                                    )                                        ⁢                                          (                                              1                                                  R                          -                          1                                                                    )                                        ⁢                                          (                                              taper                        100                                            )                                                        -                                      [                                          1                      +                                                                        (                                                      1                                                          R                              -                              1                                                                                )                                                ⁢                                                  (                                                      taper                            100                                                    )                                                                                      ]                                                  }                            }                                                          [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            6                          ]                                                                                    σ                rr                            =                            ⁢                                                1                  r                                ⁢                                  {                                                            [                                              B                        (                                                                              r                            β                                                    -                                                                                    R                                                              2                                ⁢                                β                                                                                                                    r                              β                                                                                                      )                                            ]                                        +                                                                  σ                        0                                                                    2                        ⁢                        β                                                                                                                                                                                  ⁢                              {                                                      (                                          1                                              1                        -                        v                                                              )                                    ⁢                                                            (                                              1                        -                                                  taper                          100                                                                    )                                        [                                                                                            (                                                                                                                    R                                                                  β                                  +                                  1                                                                                            -                                                              r                                                                  β                                  +                                  1                                                                                                                                                    β                              +                              1                                                                                )                                                ⁢                                                  r                                                      -                            β                                                                                              -                                                                                                                                                                                                                        ⁢                                                                  (                                                                                                            R                                                              1                                -                                β                                                                                      -                                                          r                                                              1                                -                                β                                                                                                                                          1                            -                            β                                                                          )                                            ⁢                                              r                        β                                                              ]                                    +                                                            v                      ⁡                                              (                                                  1                                                      1                            -                                                          v                              2                                                                                                      )                                                              ⁢                                          (                                              taper                        100                                            )                                        ⁢                                          (                                                                                                                                  (                                                              R                                r                                                            )                                                        β                                                    -                                                                                    (                                                              r                                R                                                            )                                                        β                                                    -                          2                                                β                                            )                                                                      }                            }                                                          [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            7                          ]            
FIG. 3 shows the radial stresses plotted as a −σα/σ0 for the two taper tension profiles. On the whole, the radial stress distribution for the hyperbolic profile has equipollence more than for the linear taper stress.
FIG. 4 shows the variation of the ERS value for the two tension profiles. In FIGS. 3 and 4, the close correlation between ERS and the radial stress can be found. As the derivative of the ERS value is low, the distribution of the radial stress is small and equal.
On the basis of the above results, it is found that the hyperbolic taper tension profile prevents intensive increment of the radial stress and promotes uniform radial stress distribution.
Camber can be expressed as the radius of the curvature in the un-tensioned condition and lying on a flat surface. Assuming linear stress distribution in the cambered web as shown in FIG. 5, the induced moment can be found in Math Figure 8:
                    M        =                              r            ×            F                    =                                                    (                                  W                  6                                )                            ⁢                              (                                                      T                    max                                    -                                      T                    min                                                  )                                      =                                                            (                                                            T                      max                                        -                                          T                      min                                                        )                                6                            ⁢              W                                                          [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            8                          ]            
From the beam theory, the curvature is shown in Math Figure 9:
                    ρ        =                  EI          M                                    [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            9                          ]            
Substituting M of Math Figure 8 into Math Figure 9 leads to the curvature model as shown in Math Figure 10:
                    ρ        =                              6            ⁢            EI                                              (                                                T                  max                                -                                  T                  min                                            )                        ⁢            W                                              [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            10                          ]            
FIG. 6 identifies the elastic behavior of the web under general movement of rollers.
In FIG. 6, the lateral deflection at a downstream roller is determined as shown in Math Equation 11 ([7] and [8]).
                              y          L                =                              2            -                          2              ⁢                              cosh                ⁡                                  (                  KL                  )                                                      +                                          sinh                ⁡                                  (                  KL                  )                                            ⁢              KL                                            ρ            ⁢                                                  ⁢                                          K                2                            ⁡                              (                                                      cosh                    ⁡                                          (                      KL                      )                                                        -                  1                                )                                                                        [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            11                          ]            
In Math Figure 11, yL is equal to telescoping error in a winding process, because the downstream roller is a wound roll. Therefore, through the correlation between lateral deflection and tension distribution, the mathematical model for telescoping can be defined as shown in Math Figure 12:
                              y          telescoping                =                              2            -                          2              ⁢                              cosh                ⁡                                  (                  KL                  )                                                      +                                          sinh                ⁡                                  (                  KL                  )                                            ·              KL                                                          [                                                                    12                    ⁢                    EI                                                                              (                                                                        F                          max                                                -                                                  F                          min                                                                    )                                        ⁢                    W                                                  ⁢                                  sin                  ⁡                                      (                                          α                      2                                        )                                                              ]                        ⁢                                          K                2                            ⁡                              (                                                      cosh                    ⁡                                          (                      KL                      )                                                        -                  1                                )                                                                        [                  Math          ⁢                                          ⁢                      FIG            .                                                  ⁢            12                          ]            
wherein K is stiffness coefficient, F is force given by web tension, and is wrap angle. FIG. 7 shows computer simulation results for the correlation between taper tension and lateral displacement for nonuniform tension distribution in the width direction of a material. FIG. 7 shows that two taper tension profiles, which show different changes in tension, are related to the occurrence of telescoping with an increase in radius.
FIG. 8 is a photograph showing a telescoping phenomenon in a prior wound roll, and FIG. 9 is a photograph showing a starring phenomenon in a prior wound roll.
As shown in FIGS. 8 and 9, the term “telescoping” refers to the widthwise displacement of material in a finally produced roll, and the term “starring” refers to star-shaped damage caused at the side of a roll due to non-uniform stress distribution. Telescoping and starring greatly influence the quality of a roll.
In a taper tension control method, which is a tension control method according to the prior art, telescoping in the beginning of rewinding can be minimized, but great radial stress occurs. In comparison with this, in a hyperbolic tension control method, telescoping in the beginning of rewinding is serious, but radial stress distribution is low.